ON MATRICES WHOSE REAL LINEAR COMBINATIONS
ARE NONSINGULAR
J. F. ADAMS, PETER D. LAX1 AND RALPH S. PHILLIPS2
Let A be either the real field R, or the complex field C, or the skew
field Q of quaternions.
Let Au A2, • ■ ■ , Ak be nXn matrices with
entries from A. Consider a typical linear combination
E"-iV^>
with real coefficients Xy; we shall say that the set {A¡} "has the property P" if such a linear combination
is nonsingular
(invertible) except when all the coefficients X> are zero.
We shall write A(ra) for the maximum number of such matrices
which form a set with the property P. We shall write Ah(») for the
maximum number of Hermitian matrices which form a set with the
property P. (Here, if A = i?, the word "Hermitian"
merely means
"symmetric"; if A = Q it is defined using the usual conjugation in Q.)
Our aim is to determine the numbers A(ra), Aff(ra).
Of course, it is possible to word the problem more invariantly.
Let W be a set of matrices which is a vector space of dimension A
over R; we will say that W "has the property P" if every nonzero
w in W is nonsingular (invertible). We now ask for the maximum
possible dimension of such a space.
In [l], the first named author has proved that i?(ra) equals the socalled Radon-Hurwitz
function, defined below. In this note we determine i?z/(ra), C(ra), Cj/(ra), Qin) and <2ff(ra) by deriving inequalities
between them and i?(ra). The elementary constructions
needed to
prove these inequalities can also be used to give a simplified description of the Radon-Hurwitz
matrices.
The study of sets of real symmetric matrices {A,} with the property P may be motivated as follows. For such a set, the system of
partial differential equations
«< = E ¿iux<
i
is a symmetric
zero in every
propagated to
To give our
hyperbolic system in which the sound speeds are nondirection. For such systems the solution energy is
infinity and a scattering theory can be developed.
results, we require the Radon-Hurwitz
numbers [2],
Received by the editors September 10, 1963.
1 Sloan Fellow.
' Sponsored by the National Science Foundation,
contract NSF-G 16434.
318
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MATRICESWHOSELINEAR COMBINATIONSARE NONSINGULAR
[3]. We set ra= (2a+l)26
with 0^c<4;
319
and b = c+4d, where a, b, c, d are integers
then we define
p(») = 2" + U.
Theorem
1. We have
R(n) = p(»),
RH(n) = p(i») + 1,
C(n) =20 + 2,
CH(n) =20+1,
Q(n) =25 + 4,
QH(n)=20+1.
The results for A = Q are included so that topologists may avoid
jumping to the conclusion that the subject is directly related to the
Bott periodicity theorems. If this were so then it would be surprising
to see the case A = Q behaving like the case A = C.
The proof of Theorem 1 will be based on a number of simple constructions, which we record as lemmas.
Lemma 1. RH(n) âCH(n) ^QH(n).
This is clear, since a matrix with entries from R may be regarded
as a matrix with entries from C, and similarly for C and Q.
Lemma 2. (a) C(n)^R(2n),
Proof. We may
coordinate »-space
it becomes a real
Thus any C-linear
(b) Q(n)^C(2n).
regard our matrices as A-linear transformations
of
A". Now by forgetting part of the structure of Cn
vector space of dimension 2» over R, i.e., an R2n.
transformation
of C" gives an i?-linear transforma-
tion of R2n. Similarly for C and Q.
Lemma 3. A(») + 1 ¿AH(2n).
Proof. Let IF be a fc-dimensional space of «X« matrices with
entries from A which has the property P. For each AEW and X£i?,
consider the following linear transformation
from A"©An to itself.
B(x, y) = (Ay + Xx, A*x - \y).
It is clear that its matrix is Hermitian, and that such 5 form a
(fe+ l)-dimensional space. We claim that this set {b} has property
P. For suppose that some 5 is singular; then there exist x, y not both
zero such that
Ay + A* = 0,
Evaluating
A*x - \y = 0.
x*Ay in two ways, we find
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320
J. F. ADAMS,P. D. LAX AND R. S. PHILLIPS
[April
X(x*x + y*y) = 0.
Hence X = 0. Thus either A or A* is singular;
so A is singular
and
.4=0. This proves the lemma.
Lemma 4. (a) CE(n) + l£C(n),
(b) QB(n)+3£Q(n).
Proof. Let Wbe a A-dimensional space of «Xra Hermitian matrices with entries from A which has the property P. Consider the
matrices
A+pI,
where A runs over W and ju runs
of A. We claim that they form a
dimension A+ l if A= C or A+3 if
matrix is singular; then there is a
over the pure imaginary elements
space with the property P and of
A = 0\ In fact, suppose that such a
nonzero x such that
Ax = — px;
arguing as is usual for the complex case, we find
—px*x = x*.4x = (—px)*x = px*x.
So ju is zero, A is singular, and thus A is zero. This proves the lemma.
Lemma 5. -R/f(ra)+7gi?(8ra).
Proof. Let W be a A-dimensional space of real symmetric matrices
with the property P. We require also the Cayley numbers K, which
form an 8-dimensional algebra over R. We can thus form the real
vector space
R" ®RK
of dimension 8ra. For each AEW and each pure imaginary pEK we
consider the following linear transformation
from Rn®RK to itself:
Bix ® y) = Ax ® y + x <g>
py.
We claim that the (A+7)-dimensional
space formed by such B has
property P. For suppose that some B is singular, and suppose, to begin with, that ju is nonzero. Then the elements 1, jtxform an i?-base
for a sub-algebra of K which we may identify with C. Now every two
elements of K generate an associative sub-algebra; in particular, K
is a left vector space over C. Choose a C-base of K; this splits Ru®rK
as the direct sum of 4 copies of R"®rC. Since B acts on each summand, it must be singular on at least one. That is, the real symmetric
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i96j]
MATRICESWHOSELINEAR COMBINATIONSARE NONSINGULAR
321
matrix A has a nonzero complex eigenvalue which is purely imaginary—a contradiction.
Hence ß must be zero and B=A®1.
Now
choose an i?-base of K; this splits Rk®rK as the direct sum of 8
copies of R". Since 5 acts on each summand, it must be singular on
at least one. That is, A must be singular; hence A=0. This completes
the proof.
Proof of Theorem 1. First we consider RH(n). If we use the fact
that R(n) =p(»), Lemmas 3 and 5 give
P»
+ 1 ^ Rn(n) Û p(8») - 7.
But using the explicit definition of p, we have
p(8») - 7 = p(» + 1.
This disposes of i?»(«).
It follows from this argument that if we have a set of p(») »X»
matrices with the property P, then by applying successively the
constructions given in the proofs of Lemma 3 (taking A = i?) and
Lemma 5, we obtain a set of p(24») 24»X24» matrices with the property P. Now the set of 1, 2, 4, and 8 matrices which express the respective actions of R, C, Q, and K on Rm, Cm, Qm, and Km lor m = 2a+l
can be used to start the induction for the different cases b=0, 1, 2,
and 3 (mod 4). This gives a slight variation of the construction of
Hurwitz and Radon [2], [3]; the iterative procedures used by these
authors require more steps and do not involve the Cayley numbers
explicitly.
Next we consider C(n). Lemmas 3 and 4(a) give
C(n) + 2 g C(2»).
Now induction
shows that
C(n) ^2b + 2,
which gives us our inequality one way. Applying Lemma 2(a)
directly only gives a good inequality for certain values of b, so we
proceed as follows. Choose e =; 0 such that b+e = 0 or 1 (mod 4). Then
by induction we have
C(«) + 2e S C(2'n);
by Lemma 2(a) we have
C(2'n) á p(2e+1»),
and by our choice of e we have
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322
J. F. ADAMS, P. D. LAX AND R. S. PHILLIPS
p(2'+1n) = 2b + 2e + 2.
This gives
C(n) g,2b + 2,
and proves the assertion made about C(n).
Lemmas 3 and 4(a) now show that
Ca(n) =26+1.
Again, Lemmas 1, 4(b) and 2(b) show that
Qu(n) - CH(n) ^ 0,
Q(n) - QH(n) è 3,
C(2») - Q(n) ^ 0.
But C(2«) —Cff(ra) = 3, so all these inequalities
completes the proof of Theorem 1.
are equalities.
This
References
1. J. F. Adams, Vectorfields on spheres,Ann. of Math. (2) 75 (1962),603-632.
2. A. Hurwitz,
Über die Komposition der quadratischen Formen, Math. Ann. 88
(1923), 1-25.
3. J. Radon,
Lineare
Scharen
orthogonalen
Matrizen,
Hamburg 1 (1922), 1-14.
Manchester University, Manchester,
New York University, and
Stanford University
England,
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Abh. Math.
Sem. Univ.